Draft version July 4, 2018Preprint typeset using LATEX style emulateapj v. 01/23/15

3D RADIATION NON-IDEAL MAGNETOHYDRODYNAMICAL SIMULATIONS OF THE INNER RIM INPROTOPLANETARY DISKS.

M. Flock1,2, S. Fromang2, N. J. Turner1, M. Benisty3

1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA2Laboratoire AIM, CEA/DSM-CNRS-Universite Paris 7, Irfu/Service d’Astrophysique, CEA-Saclay, F-91191 Gif-sur-Yvette, France and

3Universite Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France(Dated:)

Draft version July 4, 2018

ABSTRACTMany planets orbit within an AU of their stars, raising questions about their origins. Particularly puzzling arethe planets found near the silicate sublimation front. We investigate conditions near the front in the protostellardisk around a young intermediate-mass star, using the first global 3-D radiation non-ideal MHD simulationsin this context. We treat the starlight heating; the silicate grains’ sublimation and deposition at the local,time-varying temperature and density; temperature-dependent Ohmic dissipation; and various initial magneticfields.The results show magnetorotational turbulence around the sublimation front at 0.5 AU. The disk interior to 0.8AU is turbulent, with velocities exceeding 10% of the sound speed. Beyond 0.8 AU is the dead zone, coolerthan 1000 K and with turbulence orders of magnitude weaker. A local pressure maximum just inside the deadzone concentrates solid particles, favoring their growth. Over many orbits, a vortex develops at the dead zone’sinner edge, increasing the disk’s thickness locally by around 10%.We synthetically observe the results using Monte Carlo transfer calculations, finding the sublimation front isnear-infrared bright. The models with net vertical magnetic fields develop extended, magnetically-supportedatmospheres that reprocess extra starlight, raising the near-infrared flux 20%. The vortex throws a non-axisymmetric shadow on the outer disk. At wavelengths > 2 µm, the flux varies several percent on monthlytimescales. The variations are more regular when the vortex is present. The vortex is directly visible as an arcat ultraviolet through near-infrared wavelengths, given sub-AU spatial resolution.Keywords: protoplanetary disks, accretion disks, magnetohydrodynamics (MHD), radiation transfer

1. INTRODUCTION AND MOTIVATION

Our grasp of planetary systems’ origins relies on our un-derstanding of the disks of gas and dust found orbiting youngstars. Here we focus on the environment for planet forma-tion in the disks’ hot central region where temperatures ex-ceed 1000 K. Among the main processes governing the tem-peratures in this region are the sublimation and deposition ofsilicate grains, which change the opacity by orders of magni-tude (Pollack et al. 1994). Since the starlight is a major sourceof heating, the opacity has a major impact on temperatures.

A key process governing the turbulent stirring of the planet-forming materials is magneto-rotational instability (Balbus &Hawley 1991). Magnetic fields are coupled to the plasma, andthe instability converts gravitational potential energy into thekinetic and magnetic energy of turbulence (Jin 1996; Gammie1996; Miller & Stone 2000; Hirose & Turner 2011), whentemperatures are high enough for thermal ionization (Ume-bayashi & Nakano 1988; Desch & Turner 2015). The de-cline in temperature with distance from the star thus leadsto a decline in the magnetic stresses across the thermal ion-ization threshold. This means the surface density increasesacross the threshold if the inflow is in steady-state, so that themass flow rate is independent of distance. The resulting localpressure maximum can concentrate solid particles in the sizerange where their stopping time due to gas drag is comparableto the orbital period (Haghighipour & Boss 2003; Lyra et al.2008; Kretke et al. 2009; Dzyurkevich et al. 2010; Lyra &Mac Low 2012; Faure et al. 2014a), possibly allowing for in

mflock@caltech.edu

situ planet formation (Chatterjee & Tan 2014).In addition, the pressure maximum can act as a planet trap:

young planets’ inward migration under their tidal interactionwith the disk comes to a halt near the pressure peak (Massetet al. 2006; Matsumura et al. 2009; Kretke & Lin 2012; Bitschet al. 2014; Hu et al. 2015). Concentrating both pebbles andprotoplanets in a region where dynamical timescales are shorthas the potential to lead quickly to the growth of planets.

Of all young stars, perhaps the best suited for measuringthe disks’ hot central regions are the Herbig stars, whichhave masses a few times the Sun’s. Nearby examples arebright enough to be observed with near-infrared interferom-etry down to the angular scale of the disk’s inner rim (Dulle-mond & Monnier 2010; Kraus 2015) yielding maps of the sil-icate sublimation front (Benisty et al. 2011). However theseobjects’ spectra show puzzlingly large near-infrared fluxes(Hillenbrand et al. 1992; Chiang et al. 2001; Millan-Gabetet al. 2001; Meeus et al. 2001; Vinkovic et al. 2006). Radia-tion hydrostatic (Mulders & Dominik 2012) and radiation hy-drodynamic models (Flock et al. 2016) produce too little fluxat wavelengths 2-4 µm by factors up to several. Ingredientsmodelers must consider include the transfer of the starlightinto the disk, and the escape of the re-radiated infrared emis-sion; the sublimation and deposition of the dust grains thatprovide most of the opacity; and the forces supporting thedisk material against the star’s gravity (Kama et al. 2009).Models can yield near-infrared excesses closer to the observedrange if some disk material near the sublimation front is ei-ther launched into a wind (Bans & Konigl 2012) or supportedon the magnetic fields escaping from MRI turbulence within

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the disk (Turner et al. 2014a). In both pictures, the magneticforces lift some material above its hydrostatic position, in-creasing the height where the starlight is absorbed, so that abigger fraction of the stellar luminosity is reprocessed intothermal emission at distances where the emission comes outat near-infrared wavelengths. Until now, there was no globalmodeling of magneto-rotational turbulence at the inner rim,treating the dust sublimation and radiation transfer togetherwith the MHD.

In this work we present the first 3-D radiation non-idealMHD simulations of protostellar disks to include starlightheating, silicate grains’ sublimation and deposition at the lo-cal temperature and density, and Ohmic dissipation dependingon the thermal ionization. We test the results against variousobservational constraints, comparing the spectral energy dis-tributions, images, and lightcurves at different wavelengths.The models let us address several important questions: Howdoes the MHD turbulence affect the sublimation front? Whatare the consequences for the starlight-absorbing surface, thesystem’s near-infrared emission, and its time variability? Andwhat controls the dynamics at the location where the solidsare concentrated?

We describe in section 2 the radiation MHD methods andthe treatment of the dust sublimation and deposition. Sec-tion 3 deals with the results of the calculations, and section 4with the comparison against observations. We discuss the im-plications in section 5, and summarize our conclusions in sec-tion 6.

2. METHODS AND SETUP

In this section we briefly summarize the method and thesetup of the 3D radiation non-ideal MHD simulations. Therelevant equations are given in Section 2.1. The resistivitymodule which determines the magnetic field coupling param-eter is presented in Section 2.2. The initial and boundary con-ditions are presented in Sections 2.3 and 2.4. For full detailsoff the radiation transfer and the dust evaporation modules,we refer the reader to our previous works (Flock et al. 2013,2016).

2.1. Radiation non-ideal MHD equationsIn this paper, we solve the following radiation non-ideal

MHD equations in a spherical coordinate system (r, θ, φ):

∂ρ

∂t+ ∇ ·

[ρv

]= 0 , (1)

∂ρv

∂t+ ∇ ·

[ρvvT −BBT

]+ ∇Pt =−ρ∇Φ , (2)

∂E∂t

+ ∇ · [(E + Pt)v − (v ·B)B] =−ρv · ∇Φ − ∇ · F∗

−κPρc(aRT 4 − ER)−∇ ·

[(η · J ) ×B

], (3)

∂ER

∂t− ∇

cλκRρ∇ER = κPρc(aRT 4 − ER) , (4)

∂B

∂t− ∇ × (v ×B) =−∇ × (η · J ) , (5)

where ρ is the density, v is the velocity and B is the mag-netic field1. J = ∇ × B is the current density and η is thetensor magnetic resistivity. The total pressure is given by

1 The magnetic field already includes the normalization factor 1/√

4π.

Pt = P + B2/2 where the gas pressure relates to the tem-perature T through

P =ρkBTµgu

, (6)

where µg is the mean molecular weight, kB is the Boltzmannconstant and u is the atomic mass unit. T stands for the tem-peratures of the gas and dust which are assumed equal thanksto collisional exchange of thermal energy at these high gasdensities.

Introducing the gravitational constant G, the gravitationalpotential is calculated according to

Φ = GM∗/r , (7)

where M∗ is the stellar mass. E denotes the total energy andis given by the relation E = ρε + 0.5ρv2 + 0.5B2 where ρε =P/(Γ−1) is the gas internal energy, Γ being the adiabatic index.The radiation energy is denoted ER while F∗ stands for thefrequency-integrated irradiation flux. F∗ is calculated as

F∗(r) =

(R∗r

)2

σbT 4∗ e−τ∗ , (8)

with the Stefan-Boltzmann constant σb, the stellar surfacetemperature T∗ and the radius R∗ of the star. The radial opticaldepth of the irradiation flux τ∗ is calculated using the opac-ity at the stellar temperature κP(T∗) = 2100 cm2.g−1, see alsoSection 2.1 in Flock et al. (2016). κR and κP are the Rosse-land and Planck mean opacity, respectively. Both opacitiesinclude gas and dust contributions. The dust opacity is set toκP(Tev) = 700 cm2 g−1 which represents the opacity at the dustsublimation temperature per gram of dust. The gas opacity isset constant to 10−4cm2 g−1 which represents the opacity pergram of gas. For more details on the opacities we refer to ourprevious work (Flock et al. 2016). Finally, aR is the radiationconstant and c stands for the speed of light.

The gas is a mixture of hydrogen and helium with solarabundance (Decampli et al. 1978) so that µg = 2.35 andΓ = 1.42. For the typical density and temperature we con-sidered, most of the hydrogen is bound in molecular hydro-gen2. Silicates dust grains are present when the temperatureis smaller than a critical temperature noted Tev and sublimatesotherwise. As in our previous paper (Flock et al. 2016), wefollow Pollack et al. (1994) and Isella & Natta (2005) and de-termine Tev according to

Tev = 2000 K(

ρ

1g cm−3

)0.0195

. (9)

Tev is then used to calculate the local dust density ρd, assum-ing perfect mixing of dust and gas (see Eq.(10) in Flock et al.2016).

2.2. ResistivityWe implemented a simple treatment of the resistivity in or-

der to model the ionization transition at 1000 K (Umebayashi& Nakano 1988; Desch & Turner 2015). To do so, we set

η =c2

s

ΩRem, (10)

2 We have checked that for ρ < 10−14 g.cm−3 and T < 1500K over 50 %of the hydrogen is bound in H2.

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 3

M 10−8 M/yearStellar parameter T∗ = 10000 K, R∗ = 2.5 R, M∗ = 2.5 MOpacity κP(T∗) = 2100 cm2 g−1

κP(Tev) = 700 cm2 g−1

κgas = 10−4 cm2 g−1

Dust to gas mass ratio f0 = 0.01Cell aspect ratio R∆θ/∆R : R∆Φ/∆R ∼ 1.1 : 1.2Rin − Rout : Z/R 0.3-3AU : ±0.36Nr × Nθ 896 x128RMHD P0 4 Φmax = 0.4, Nφ = 128, runtime 300 orbitsRMHD P1 6 Φmax = 1.6, Nφ = 512, runtime 150 orbitsRMHD P0 4 BZ Φmax = 0.4, Nφ = 128, runtime 70 orbits

Table 1Model parameter for the radiation MHD model RMHD P0 4, RMHD P1 6 and

model RMHD P0 4 BZ.

where the magnetic Reynolds number Rem dependence on thetemperature is given by

Rem = 5 × 104(1 − tanh

(1000 − T

25

)). (11)

The asymptotic value of the magnetic Reynolds numberRem = 105 is used for temperatures above 1000 K. This up-per limit is high enough to obtain sustained MRI turbulenceclosely resembling the ideal-MHD limit (Flock et al. 2012b).For temperatures below 1000 K the Reynolds number de-creases and we set the lower limit of the magnetic Reynoldsnumber to ReDZ

m = 1. Such a value is low enough to ensurethe damping of the MRI inside the dead-zone (Elsasser num-ber 1).

2.3. Initial conditionsIn order to initialize the simulations presented here, we used

the final snapshot of the axisymmetric radiation viscous 2.5D3

hydrodynamical simulation RMHD 1e-8 of Flock et al. (2016),for which the flow has reached a steady-state characterized bya uniform accretion rate of M = 10−8 solar mass per year.We refer the reader to Flock et al. (2016) for more details onthat particular simulation, and only show here the resulting2D distribution of ρ, ρd and T in the disk meridional plane(Fig.1). These 2D fields are extended in the azimuthal direc-tion to cover the range [0, 0.4] radian and [0, 1.6] radian forthe models RMHD P0 4 and RMHD P1 6, respectively.

To trigger the MRI, we investigate two different configura-tions for the magnetic field geometry at the beginning of thesimulations. First, a random zero-net flux magnetic field isused for the models RMHD P0 4 and RMHD P1 6. A snapshotof the initial magnetic field is shown in Fig. 1 (bottom panel).For model RMHD P0 4 BZ, we also added a vertical net fluxmagnetic field. In Appendix B, we detail the procedure weused to generate the magnetic vector potential in both cases.The naming convention of the radiation MHD (RMHD) mod-els includes the size of the azimuthal domain given in radians(e.g. P0 4 for Φmax = 0.4) and if a vertical magnetic field isincluded, BZ. The model parameters are summarized in Ta-ble 1.

2.4. Boundary conditions and buffer zonesIn the radial direction we use zero gradient conditions for all

variables while vr is set to enforce vanishing mass inflow. In

3 2.5D represents the use of 2 dimensions (r,θ) and 3 components (r,θ.φ)for the velocity and magnetic field vectors.

Figure 1. Initial profile in the R-Z/R plane for the gas density (top), the dustdensity (second row) and the temperature (third row). Those are snapshotstaken from 2D radiation viscous hydrodynamical simulations in steady-state.Bottom: Meridional slice of the initial Bθ magnetic field component, gener-ated from the vector potential Ar .

the meridional direction we extrapolate the logarithmic den-sity and the temperature in the ghost cells. The ghost cellsare a set of additional cells at the domain boundary whichprovide the boundary values for the integration. For the ve-locity vθ we set a zero mass inflow condition. The azimuthalboundaries are periodic. In addition, we use a buffer zoneat the radial inner boundary over the radial range [0.3, 0.35]AU to avoid effects arising from the presence of the bound-ary. In this zone we damp the radial and vertical velocitiesand increase linearly the magnetic resistivity in order to reacha magnetic Reynolds number of 10 at the location of the in-ner radial boundary. In addition, we used the same modifiedgravitational potential inside the buffer zone as used in the2D radiation hydrodynamical simulations (see Appendix E inFlock et al. 2016). This last modification affects the regionwith |Z/R| > 0.1 and R < 0.35AU, and manifests itself asa layer with temperatures slightly larger than expected (seewhite zone in Fig. 1, third panel). R and Z represent here thecylindrical coordinates. This buffer zone is excluded from theanalyses presented below.

2.5. Diagnostics

4 Flock et al.

Figure 2. Time evolution of the stress to pressure ratio α over radius formodel RMHD P0 4. Midplane positions are shown for (from left to right) theinner rim (green dashed line), the 900 K temperature contour (yellow dashedline), the inner dead-zone edge with Λ = 1.0 (yellow solid line) and the dustconcentration radius (blue dashed line).

We determine the strength of the turbulence by calculatingthe stress to pressure ratio α:

α =

∫ρ

(Trφ

P +Mrφ

P

)dV∫

ρdV=

∫ρ

(ρv′φv′r

P −BφBr

P

)dV∫

ρdV, (12)

which is the sum of the Reynolds stress Trφ and Maxwellstress Mrφ

4. Radial profiles of α are obtained by integrationalong θ and φ. 2-D profiles are obtained by integration alongφ.

We note that the time units are inner orbits, which alwaysrefer to the orbital time at 0.3 AU.

3. RESULTS

We focus our analysis on the dynamics of three distinct re-gions of the disk. The first is the inner rim, which we define asthe irradiation τ∗ = 1 line. The second is the inner edge of thedead-zone, defined as the region where the Elsasser numberΛ = v2

a/(ηΩ) (va being the Alfven velocity) is smaller thanunity. MRI turbulence is damped when Λ is less than unity(Sano & Stone 2002). The third is the dust concentrationradius, and is defined as the location when pressure reachesa maximum and where the gas azimuthal velocity is exactlyKeplerian. The gas is rotating with super(sub)-Keplerian ve-locities inward (outward) of that location. We will focus onthose three regions in the following analysis and mark theirpositions in Figs. 2 to 7.

We first investigate the detailed disk dynamics using the re-sults of model RMHD P0 4 in Sections 3.1 and 3.2. The re-sults of model RMHD P0 4 BZ are presented in Section 3.3.The question of whether large-scale and long-term non-axisymmetric perturbations arise in the simulations is investi-gated using model RMHD P1 6 in Section 3.4.

4 We compared the mass weighted integral for the α parameter with theclassical definition α =

∫(Trφ + Mrφ)dV/

∫PdV . The radial profiles are

almost identical with maximum relative deviations of around 5 % close to theinner rim.

3.1. Time evolutionFig. 2 shows the time variations of α(R) for modelRMHD P0 4. The immediate result is that the model quicklyreaches a quasi-steady state. As a result of the MRI, the flowis turbulent in the inner disk (r < 0.8 AU) during the entiresimulation, with typical values of α ∼ 0.03. Such a rate forthe flux of angular momentum is in agreement with previouslocally isothermal global ideal MHD simulations (Fromang &Nelson 2006; Flock et al. 2011; Parkin & Bicknell 2013) andglobal radiation ideal MHD simulations (Flock et al. 2013).

The inner rim (τ∗ = 1) in the midplane is located at around0.47 AU inside this highly turbulent region (green dashedline). Its position remains almost unaffected by the turbulencebecause the optical depth to the star is the line integral througha large number of uncorrelated turbulent density fluctuations.The dead-zone inner edge is located at around 0.8 AU (yellowsolid line). At this position, the Elsasser number Λ drops be-low unity. In our model, the dead-zone inner edge is also closeto the temperature contour T ∼ 900 K (yellow dashed line)and corresponds to the location where the magnetic Reynoldsnumber drops below ∼ 100. At this position, the value of αdrops by several orders of magnitude. We note that the ex-act radial position of the dead-zone inner edge shows varia-tions of around 2 to 3 disk scale heights in the radial directionwith time (with typical displacement of the order of ∆R ∼ 0.1AU). This is due to temperature variations associated with theturbulence. Finally, the dust concentration radius is locatedinside the dead-zone at around 1 AU (blue dashed line) anddisplays little sign of radial variations.

We now take a deeper look at the time evolution of the diskstructure. Fig. 3 shows the time evolution of the surface den-sity and three selected temperature contours. For the simula-tion runtime the disk remains stable. Overplotted temperaturecontours in steps of 400 K show the temperature variationsover time at different disk locations. Close to the inner rim,the 1200 K contour show only small radial fluctuations. The800 K and 400 K contours behind the inner rim move slightlyradially inward by around 0.1 AU in the first 40 local orbitsdue to the extended height of the dust rim by magnetic fields,which causes a steeper temperature drop in the shadow behindthe rim. Fig. 4 shows the time averaged radial temperatureprofile of model RMHD P0 4 overplotting the standard devia-tions. The temperature fluctuations remain small. The max-imal deviations are around 10-30 K and appear due to varia-tions of the rim surface and so the grazing angle variations ofthe irradiation.

To summarize, the disk structure for model RMHD P0 4 re-mains stable over the simulation time. The model shows aquasi steady-state on dynamical timescales. However we em-phasize that this model is not in inflow equilibrium on thelonger accretion flow timescale. The initial surface densityprofile was calculated assuming αDZ = 10−3 for temperaturesbelow 1000 K. In our 3D models we find values of α whichare orders of magnitude smaller. In such a case the surfacedensity should be much higher to balance the drop in accre-tion stress. In the discussion section we explore this pointfurther.

3.2. Time averaged results: Zero-net flux modelThe time averaged meridional distribution of the stress to

pressure ratio for model RMHD P0 4 is plotted in Fig. 5, alongwith the positions of the inner rim and the dead-zone inneredge (green and yellow dashed lines, respectively). The plot

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 5

Figure 3. Time evolution of the surface density over radius for modelRMHD P0 4. Temperature contours are shown for 1200 K (green line), 800 K(yellow line), and 400 K (blue line).

Figure 4. Time averaged and meridionally integrated radial profiles of thetemperature for model RMHD P0 4. Time average is done between 20 and 300inner orbits. The blue shadow shows the standard deviations in the profile.Vertical lines show the midplane positions of the inner rim (green dashedline), the inner dead-zone edge (yellow line) and the dust concentration radius(blue dashed line).

Figure 5. 2D profile of the stress to pressure ratio α in the R-Z/R plane formodel RMHD P0 4. Time average is done between 20 and 300 inner orbits.Overplotted are the contours for the inner rim (green dashed line) and theinner dead-zone edge (yellow line) (Λ = 1.0).

shows that the entire inner rim surface is embedded in the tur-bulent region of the disk. The dead-zone edge is located atR ∼ 0.8 AU. Its shape is roughly vertical around the disk mid-plane (|Z/R| < 0.12) while in the upper disk layers it curvesradially outward, consistent with the fact that the disk surfaceis radiatively heated by the central star. In agreement with theresults described above, turbulent activity quickly drops to avery small value inside the dead-zone.

The time averaged and meridionally integrated 1D pro-

Figure 6. Time averaged and meridionally integrated radial profiles formodel RMHD P0 4, including the stress to pressure ratio α (top), the turbulentvelocity (middle) and the turbulent magnetic field (bottom). Time average isdone between 20 and 300 inner orbits. Vertical lines show the midplane po-sitions of the inner rim (black dotted line), the inner dead-zone edge (yellowline) and the dust concentration radius (blue dashed line).

file of the stress to pressure ratio α is shown in Fig. 6 (toppanel). The plot features a plateau with a constant value ofα = 0.03 over the entire inner disk, without any noticeablechange across the dust rim. In agreement with previously pub-lished simulations performed in the ideal MHD limit (Bran-denburg et al. 1995; Miller & Stone 2000; Simon et al. 2009;Davis et al. 2010), the Maxwell stress is around 3 times largerthan the Reynolds stress. At the dead-zone inner edge, theMaxwell stress drops sharply by roughly four orders of mag-nitude. By contrast, the Reynolds stress decrease is more shal-low, so that it dominates the total stress in a small region be-tween 0.8 and 1.1 AU which corresponds to roughly 10 diskscale heights. Over that region, the total stress takes valuesthat range between α = 10−4 to 10−6 times the pressure. TheMaxwell stress amounts to about 10−6 and is governed bymean magnetic fields which diffuse into the dead-zone. Themean Maxwell stress is computed from the volume averagedfields, see red dashed curve on the top panel. The relativelylarge Reynolds stress is mainly due to density waves whichare excited in the inner disk turbulence (Heinemann & Pa-paloizou 2009a). Previous global MHD simulations of the in-

6 Flock et al.

ner dead-zone edge have found similar amounts of Reynoldsstress close to the dead-zone inner edge (Dzyurkevich et al.2010; Faure et al. 2014b). These density waves propagate intothe dead-zone and are quickly damped after around 10 scaleheights. In Appendix C we review and discuss in more detailthe damping mechanism. Finally, outside of 1.1 AU, α dropsfurther to values of about 10−7.

The radial profile of the (mass weighted) turbulent velocityfluctuations shown in Fig. 6 (middle panel) displays similar-ities with the radial profile of α. It plateaus inside 0.8 AU,with values between 300 and 400 meters per second that cor-respond to several tens of percent of the sound speed. At therim position, there is a small decrease that we trace to the sud-den temperature decrease and the associated drop in the soundspeed (see red dotted line in Fig. 6, middle panel). This isnot surprising since the value of α remains roughly constantacross the inner rim. At the dead-zone inner edge (r ∼ 0.8AU), the turbulent velocities quickly drop by several ordersof magnitude so that, at the position of the dust concentra-tion radius (r ∼ 1 AU), the turbulent velocity fluctuations arereduced to values around 10 m s−1.

The radial profiles of the turbulent and mean magnetic fieldgiven in Fig. 6 (bottom panel) were calculated by perform-ing a simple average over one scale height above and be-low the midplane. Inside the MRI active region, the turbu-lent magnetic fields show a power law radial profile with aslope of about −3/2 and a strength of one Gauss at 0.6 AU.This profile is slightly steeper than the r−1 radial profile foundin toroidal net-flux global 3D isothermal simulations (Flocket al. 2011) and also recently again by Suzuki & Inutsuka(2014) in global simulations with a vertical net flux field. Theradial profile of the turbulent and mean magnetic field (thelatter being dominated by the toroidal component) both fol-low the profile that would be expected in a gas characterizedby a constant β value, where β = 2P/B2 is the ratio betweengas and magnetic pressure (see red dotted line which corre-sponds to β = 100). At the dead-zone inner edge, the tur-bulent field strength quickly drops leaving a dominant meanmagnetic field inside the dead-zone. The plot shows that themean magnetic field diffuses outward in the dead-zone fromits inner edge at 0.8 AU.

3.3. Stronger turbulence: the net flux modelIn this section we present the results of modelRMHD P0 4 BZ. It is intended to simulate the conditionsin a protoplanetary disk which is under the influence of avertical magnetic field. Model RMHD P0 4 BZ was computedby restarting model RMHD P0 4 after 210 inner orbits, addinga uniform vertical magnetic field (see Appendix B for details)whose strength is such that β = 3.5 × 104 at 1 AU in thedisk midplane. We find that the radial locations of the innerrim, the dead-zone edge and the dust concentration are onlyweakly modified compared to model RMHD P0 4.

Fig. 7 (top panel) plots the time averaged and meridionallyintegrated radial profile of the stress to pressure ratio. Timeaveraging is done between 20 and 70 inner orbits. The modelRMHD P0 4 BZ shows a substantially increased turbulent ac-tivity compared to model RMHD P0 4, with a high plateau ofα ∼ 0.1 in the inner disk. Such high α values are expectedin the ideal MHD limit when the disk is threaded by a ver-tical net flux magnetic field (Bai & Stone 2013). The meanMaxwell stress is large even in the MRI active region and ac-counts for about half of the angular momentum transport. Itdrops by one order of magnitude only at the dead-zone inner

Figure 7. Time averaged and vertical integrated radial profiles for modelRMHD P0 4 BZ, showing the stress to pressure ratio α (top), the turbulent ve-locity (middle) and the turbulent magnetic field (bottom). Time average isdone between 20 and 70 inner orbits. Vertical lines show midplane positionsof the inner rim (black dotted line), the inner dead-zone edge (yellow line)and the dust concentration radius (blue dashed line).

edge (see red dash-dotted line) and dominates the total stressin the bulk of the dead-zone, with a typical α of order 10−3.Such large values generated by axisymmetric magnetic fieldshave also been found in local box simulations that include aOhmic dead-zone only (see model V1 by Turner et al. (2007),model X1d by Okuzumi & Hirose (2011) or model D1-NVFbby Gressel et al. (2012)).

Next the middle panel of Fig. 7 shows the radial profile ofthe time averaged and mass weighted turbulent velocity fluc-tuations. Compared to RMHD P0 4, vrms is increased by a fac-tor of two to three in the inner disk. Finally, the time averagedturbulent and mean magnetic fields are shown in Fig. 7 (bot-tom panel). Again, in the MRI active region, we find an in-crease by a factor two to three compared to model RMHD P0 4.Their radial slopes, however, remain similar in this region.The biggest difference is the presence of a large mean field inthe whole domain (compare black dashed lines in the bottompanel of Fig. 6 and Fig. 7). This is particularly true in thedead-zone, where the mean field in model RMHD P0 4 BZ isalmost two orders of magnitudes larger than the typical valuewe found in model RMHD P0 4. Finally, we caution that the

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 7

Figure 8. 3D volume rendering of the dust density (left) and the toroidal magnetic field (right) for model RMHD P1 6 after 50 inner orbits. On the left, the locationof the inner rim (0.47 AU), the dead-zone edge (0.8 AU) and the dust concentration radius (1 AU) are annotated. On the right, the domain’s top half is cut awayto show the magnetic field on the midplane.

Figure 9. Azimuthal density variations in the largest azimuthal mode overtime in model RMHD P1 6 at three different location inside the disk. After 20local orbits (see upper x-axis), the non-axisymmetric perturbation close to thedead-zone inner edge grows by two order of magnitude, reaching order unityin the upper layers (red dashed line).

total duration of the simulation for model RMHD P0 4 BZ re-mains fairly small. We refer the reader to the discussion sec-tion for more details.

3.4. Long lasting non-axisymmetric perturbationsIn this section, we investigate the potential growth of

non-axisymmetric structures. This is done by using modelRMHD P1 6, which is similar to model RMHD P0 4 but featuresa larger azimuthal extent. The initial conditions for this sim-ulation are generated using a snapshot of the flow in modelRMHD P0 4 after 150 inner orbits and periodically repeatingthe azimuthal domain four times. Velocity perturbations ofthe order 10−4cs are applied cellwise to each component tobreak the symmetry. Model RMHD P1 6 quickly reaches anew turbulent state, albeit with statistical properties similarto model RMHD P0 4 (see Appendix A). The 3D rendering ofthe dust density after 50 inner orbits is shown in Fig. 8 (leftpanel). The plot confirms that dust particles are found be-tween the rim and the dead zone inner edge in a highly tur-bulent environment. As discussed in the previous sections,the dust density increases sharply at the dead zone inner edge,following a similar increase in the gas surface density that isdue to the drop of the accretion stress (Flock et al. 2016). Theright panel shows the tangled structure of the turbulent mag-netic field. It is dominated by the toroidal component whichreaches amplitudes of several Gauss.

We next compute discrete Fourier transforms of the densityalong azimuth. We focus on three different locations: (1) the

midplane at 0.5 AU, which is fully turbulent, (2) the upperlayer close to the inner rim surface and the dead-zone edge(R=0.83 AU and Z/R = 0.13), and (3) the midplane at 0.83AU. The results are summarized in Fig. 9. Initially, the largescale density variations are weak. They amount to a few per-cent in both the turbulent midplane at R = 0.5 AU and in thedisk upper layers at R = 0.83 AU (see black solid and reddashed lines). They are even smaller in the disk midplane atR = 0.83 AU (i.e. at the dead-zone edge) where they onlyreach value of ∼ 10−3 (red dotted line). In the MRI activeregion, these density perturbations do not grow for 60 localorbital periods (∼ 129 inner orbits). However, there is a clearincrease by about two orders of magnitude at the location ofthe dead-zone edge (both red lines). In particular, the rela-tive perturbations reach order unity in the disk upper layers atR = 0.83 AU. The presence of a sharp surface density changeat that location and the growth timescale of ∼ 20 local or-bits both suggest the Rossby wave instability (RWI) (Lovelaceet al. 1999; Meheut et al. 2013). This is confirmed by the ap-pearance of a localized vortex (not shown) characterized by amidplane relative vorticity of about (∇ × v)z/Ω ≈ −0.3 in thevortex core. Similar values have been reported in the literaturefor vortices produced by the RWI (Lyra & Mac Low 2012;Meheut et al. 2013; Flock et al. 2015). Fig. 9 indicates thatthe vortex is not growing anymore after 20 local orbits. Re-garding the expected lifetime of the vortex we can only makepredictions based on its shape. The vortex has an extent of∼ 2H in radius and around ∼ 22H in azimuth (H/R ∼ 0.04at the vortex location). Such an elongated vortex is knownto be vulnerable to elliptical instabilities (Lesur & Papaloizou2009), which will limit its subsequent growth. An alternatingvortex growth and destruction, as it was found in (Flock et al.2015), could be also possible.

The vortex appears in Fig. 10 (top panel) in the snapshotof the dust density after 150 inner orbits as a clear non-axisymmetric density maximum. The middle and bottom pan-els of Fig. 10 show the dust density in the R-Z/R plane insideand outside the vortex (along the lines labeled “A” and “B”on the top panel, respectively). Inside the overdensity, the ir-radiation optical depth unity line (or, equivalently, the heightof the rim) is increased vertically (see the dotted lines on themiddle and bottom panels of Fig. 10): at R = 0.85 AU, wemeasured (Z/R)τ∗=1 = 0.17 for cut “A”, and (Z/R)τ∗=1 = 0.14for cut “B”. As will be further discussed in section 4.3, sucha difference in height is enough to create an extended shadow

8 Flock et al.

Figure 10. Cross-sections through the dust density in different orientations.The face-on view (top) is cut at the height Z/R=0.13. The red dashed lineshows the positions of the two slices with the vortex (middle) and without(bottom). The black lines indicate optical depth unity for the irradiation (dot-ted line) and for the thermal emission (solid line).

on the disk beyond.

4. OBSERVATIONAL CONSTRAINTS

In this section we post-process our models with Monte-Carlo radiation transfer tools in order to translate the resultsdescribed in the previous sections into observational con-straints. This is done using the Monte Carlo radiative trans-fer code RADMC3D (Dullemond 2012) for which we use thesame parameters as in Flock et al. (2016). For more details onthe post-processing and the RADMC configuration we use,we refer the reader to Appendix D. We first focus on the disk

Figure 11. SED calculated for the 3D radiative MHD models RMHD P1 6(black dotted line) and model RMHD P0 4 BZ (red dashed line), normalizedover the SED from the initial 2D radiation HD model.

Figure 12. Space (azimuthally) and time (50 inner orbits) averaged plasmaβ profile in the R-Z/R plane for model RMHD P0 4 BZ. The line of plasmaβ unity (red solid line) and the τ∗ = 1 line (green dashed line) are over-plotted. The solid yellow line shows the position of the peak emission at2.2µm assuming a face-on orientation. The yellow dotted lines and the shad-ing demonstrates the spatial extent where 50% of the total emission at 2.2µmis coming from.

Figure 13. Radial cut from a face-on synthetic image calculated at 2.2µm forthe model RMHD P1 6 and RMHD P0 4 BZ.

spectral energy distribution (SED) in Section 4.1 and nextcompute synthetic images in Section 4.2. Then, we discussthe time variability associated with the disk dynamics in Sec-tion 4.3.

4.1. SED

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 9

First, we determine the SED of the models RMHD P1 6 andRMHD P0 4 BZ and compare them with the initial radiationHD models. The system is seen inclined 45 from face-onand the azimuthal domain is repeated to cover the entire 2π az-imuthal range. We calculate the flux at seven individual wave-lengths between 1 and 7 µm as it is the relevant wavelengthrange for our domain size and temperature range. The re-sults are plotted in Fig. 11. The 3D radiation non-ideal MHDmodel RMHD P1 6 is very close to the 2D radiation hydrody-namical model, with a modest increase of the emission around2µm of 5% due to the weak magnetized corona. By contrast,model RMHD P0 4 BZ, for which the magnetic activity is muchstronger, shows a significant increase of about 20% comparedto the hydrodynamical model.

A discussion of the origin of the emission arising at dif-ferent wavelengths is enlightening to understand this differ-ence. Most of the NIR emission is thermal in origin andcomes from the surface located where Z/R < 0.05 and R <0.6 AU. At these locations, the magnetic field is weak anddoes not alter the density distribution: the area of the emit-ting region is unchanged. This can be illustrated with the helpof the quantity β, which indicates the importance of gas rel-ative to magnetic pressure. Fig. 12 shows the distribution ofβ in the R-Z/R plane for model RMHD P0 4 BZ, averaged intime and in azimuth. The upper layers are magnetically dom-inated (β ≤ 1) while the midplane region remains gas pres-sure dominated (β ≥ 1). The equipartition line (β = 1) staysabove the rim surface (τ∗ = 1) for R < 0.6 AU. Most ofthe J and K band NIR emission is coming from a narrow re-gion: the solid yellow line in Fig. 12 marks the location of thepeak emission (solid) and the area between the dashed yel-low lines corresponds to the location where 50% of total fluxat 2.2µm is emitted (at face-on orientation). Although it isshown here for model RMHD P0 4 BZ, this result is similar formodel RMHD P1 6. Fig. 13 helps to understand the increasedNIR emission in model RMHD P0 4 BZ: it shows a radial cutfrom the synthetic image at 2.2µm for both models. The re-gions outward from 0.6 AU mainly contribute to the increasedemission in model RMHD P0 4 BZ. As seen in Fig. 12, themagnetic pressure is stronger than the gas pressure at the po-sition of τ∗ = 1 which results in increased magnetic support,and thereby a shallower density profile in the disk upper lay-ers. By contrast, model RMHD P1 6 presents weaker magneticactivity and a thinner magnetically supported corona whichleads to the SED profile being closer to the hydrodynamicalmodel. Overall, these results suggest that magnetic field areable to increase the emission between 2 to 3 µm by 5 to 20%.We discuss the possibility of obtaining an even higher NIRexcess in section 5.

4.2. Synthetic imagesSynthetic images are shown in Fig. 14 corresponding to the

radiation hydrodynamical model (top panel) and the global3D radiation non-ideal MHD models (bottom three panels).For all cases, the images cover a region which is approxi-mately 2 AU wide. The fluxes at 1.25, 2.2, and 4.8 µm fromthe blue, green and red channels on a shared linear color scalenormalized over the maximum intensity at 4.8 µm. In the syn-thetic images of the 3D models computed at early times dur-ing the simulations (second and third panels), small turbulentstructures can be identified, especially in the uppermost lay-ers of the near-infrared emitting region. Model RMHD P0 4 BZ(second panel) shows a slightly narrower and brighter ringcompared to model RMHD P1 6 (third panel). This is because

Figure 14. Synthetic images of the initial radiation HD model (top) and theglobal 3D radiation non-ideal MHD models RMHD P0 4 BZ (second) after 50inner orbits, and model RMHD P1 6 (third and bottom) after 50 and 150 in-ner orbits. The field of view is 2 AU wide, and the system is inclined 60from face-on. The blue, green, and red channels in each image correspondto wavelengths 1.25, 2.2, and 4.8 µm, corresponding to J, K, and M bands,respectively. The plot shows linear intensity. Normalization is done over themaximum intensity at 4.8 µm. We note that 2π coverage is obtained by repet-itively extending the azimuthal domain, which leads to the repeating spiralstructure for model RMHD P1 6 (bottom).

the rim surface is slightly steeper compared to the other mod-els (see also the τ∗ = 1 line in Fig. 12) owing to the mag-netic support. At later time during the evolution of modelRMHD P1 6, the effect of the vortex becomes visible in the syn-thetic image of the NIR emission (bottom panel in Fig. 14) asa spiral pattern visible in the M band emission. We note thatthe m = 4 pattern comes from duplicating the simulation do-main in the azimuthal direction before viewing the snapshot.

10 Flock et al.

Figure 15. Synthetic image at 0.3 µm after 150 inner orbits for modelRMHD P1 6. The system is inclined by 45. We note that the 2π coverageis obtained by repetitively extending the azimuthal domain, which leads tothe m=4 pattern.

Global 2π hydrodynamical calculations indicate such multi-ple RWI vortices merge leaving a single m = 1 pattern (Lyra& Mac Low 2012; Meheut et al. 2012).

Using the final snapshot of model RMHD P1 6, we also com-puted the synthetic image at 0.3 µm in Fig. 15. At this wave-length, most of the surface brightness is due to star light scat-tered from the disk surface. The strongest scattering happensat the rim, producing the ring structure outward of 0.5 AU.The spiral structure due to the vortex is also visible. At largerradii (R > 1 AU), the intensity drops by orders of magnitudeand a shadow is visible. The vortex increases locally the in-ner rim height, throwing a non-axisymmetric shadow onto theouter disk. Almost all the scattered 0.3-µm flux of the systemcomes from the inner rim. As recently discussed by Dong(2015), the shadow structure shows a smooth profile, withoutany sharp transitions. Although we do not expect such a struc-ture to be observed given the spatial resolutions that can bereached with current telescope facilities, such shadowing byan inner vortex might affect the disk temperature (and there-fore its dynamical response) through its impacts on heatingand cooling.

4.3. VariabilityWe now investigate potential variability using the results

of model RMHD P1 6. We focus on two aspects. The first isthe variability of the NIR emission caused by variations ofthe rim surface and shape. The second is related to the star’soccultation by the inner rim.

4.3.1. NIR intensity variability

We start with the variability of the NIR emission, which wecalculate for a system viewed with an inclination of 45. Thelightcurve is plotted in Fig. 16 for different wavelengths. At1.25µm, the variations are smaller than 1%. At this wave-length, most of the emission is coming from the opticallythin and hot material close to the midplane position of the in-ner rim. Larger variations become visible at the wavelengths2.2µm and 4.8µm, reaching up to ±5% in relative amplitude.At longer wavelengths, e.g. at 7µm, the fluctuations amplitudedecreases again. Fig. 16 also shows clearly that the variations

Figure 16. Top: Intensity over time for 1.25µm (blue), 2.2µm (green), 4.8µm(red) and 7.0µm (black) at 45.0 inclination from model RMHD P1 6. Thelegend lists the normalizing intensities in units of 10−18 erg cm−2 s−1 Hz−1.Bottom: Time evolution of the mean toroidal magnetic field, determined at aradius of 0.5 AU and a height of Z/R=0.05. Two years roughly correspondsto 10 Keplerian orbits at 0.5 AU.

at different wavelengths are correlated.The NIR emission exhibits two different types of varia-

tions. During the early evolution (0-10 years), Fig. 16 showslow frequency variations with a period of two years (whichroughly corresponds to 10 Keplerian orbits at 0.5 AU). Thesevariations could be connected to the oscillations of the meantoroidal magnetic field which happen to display a similartimescale of 10 orbits (see Fig. 16, bottom panel, where weplot the mean toroidal magnetic field at a radius of 0.5 AUand a height of Z/R=0.05). Oscillations of the mean toroidalmagnetic field such as reported here are known to be a robustoutcome of MRI-driven MHD turbulence in disks (Stone et al.1996; Miller & Stone 2000; Lesur & Ogilvie 2008; Gressel2010; Simon et al. 2011; Flock et al. 2012a). Our results thuspotentially suggest an indirect signature of this dynamical fea-ture in the NIR emission variability of Herbig stars. At latertime during the simulation (t > 10 years), the frequency ofthe variability changes and the NIR emission starts to displaymonthly time variations. These are due to the vortex rotat-ing around the star and depend on the combination of vortexazimuthal angle and disk inclination.

4.3.2. Stellar occultation by the inner rim

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 11

Figure 17. Top: Intensity over time at 0.3 µm and at 81.1 inclination of theglobal 3D radiation non-ideal MHD model RMHD P1 6. The dips are causeddue to variations in the dust density at the inner rim surface. Bottom: Finersampling of the intensity over Φ angle at 0.3 µm and at 81.1 inclinationusing snapshots for two different times. The upper time axis corresponds tothe rotation at 0.9 AU, the position of the occulting rim.

Finally, we investigate the variability associated with theoccultation of the star due to variations, both in space andtime, of the inner rim height. Such height variations will pro-duce a time-varying absorption of the stellar light which westudy here. To do so, we calculate a time series of the relativeintensity (i.e., normalized by the mean intensity) received byan observer at 0.3 µm when viewing the disk at an inclinationθ = 81.1. We chose that particular inclination because thisis the value of θ for which the variability is the largest. Thewavelength at 0.3 µm was chosen to represent the variationsat the peak emission flux for this type of star.

The time evolution of the 0.3 µm intensity is shown inFig. 17 (top panel), with a time sampling of 0.1 years (∼ 1/6of an orbital period at 1 AU for this type of star). The am-plitude of the variability is initially (i.e at times t < 10 yrs)of the order of 50% and displays an irregular pattern. It iscaused by the turbulent motions at the rim surface. In addi-tion, the mean field oscillations reported in Fig. 16 are alsoable to increase the density along the line-of-sight for a given

Figure 18. Evolution of the surface density over time shown at the dead-zoneinner edge (0.85 AU) for model RMHD P0 4.

time. At later times during the evolution (t > 10 yrs), thevortex causes the variations to become more periodic with in-tensity fluctuations up to an order of magnitude. To obtain afiner sampling of both occultation patterns, we make two ad-ditional series of Monte Carlo radiative transfer calculations,one for a representative snapshot from the pre-vortex stageof the disk’s evolution, and the other for the later stage. Foreach, we simulate the time changes by systematically increas-ing the azimuthal angle from which we view the disk. Theresults are shown in Fig. 17 (bottom panel). At t = 4.5 yrs(dashed curve), we find variations of the order of 20% with acorresponding timescale of about ten days. They are due toturbulent structures located at the rim surface and correspondto the high frequency fluctuations seen at early times on thetop panel of Fig. 17. At t = 14.5 yrs (solid curve), we recoverthe periodic modulation of the intensity, with an amplitude ofroughly one order of magnitude, seen at late times on the toppanel of Fig. 17. As discussed previously (see the variationsof the τ∗ = 1 line of Fig. 10), this is due to the rim heightlocally increasing by about 10% at the location of the vortex.

5. DISCUSSION

In this section, we discuss some limitations of our model-ing.

5.1. Zero-net-flux modelsIn section 3.1 we have shown that the disk structure re-

mains stable for the simulation runtime. However, the ac-cretion stress inside the dead zone which is found for modelRMHD P0 4 and model RMHD P1 6 is orders of magnitudelower than what was assumed to generate the initial condi-tions. The value of the surface density inside the dead-zoneshould therefore be taken with care. Fig. 18 shows the sur-face density evolution over time at the dead-zone inner edge at0.85 AU for model RMHD P0 4. There we observe the fastestsurface density variation. The surface density at the dead-zone inner edge increases around 10% over the runtime. Overthe accretion flow timescale we would expect a gradual in-crease of the surface density in the dead zone. Spanning suchtimescales is not feasible in 3D simulations.

5.2. Vertical net flux modelModel RMHD P0 4 BZ shows a much higher accretion stress

in the dead zone, close to the value assumed in computing

12 Flock et al.

the initial conditions. However, one should be careful in in-terpreting the magnetic field strength and geometry at largeradii in this model given the short simulation time. The timeaverage of 50 inner orbits represents only ∼ 1.6 outer orbitsand the high stress we observe in the dead-zone might only bedue to the initial conditions. Model RMHD P0 4 BZ is also lim-ited by severe mass loss associated with a strong magneticallydriven outflow. Local and global simulations by Fromanget al. (2013) and Suzuki et al. (2010) suggest that the surfacedensity would decrease on timescales of ∼ 100 local orbits asa result. This will eventually also affect the optical depth andthe thermal structure at the rim, setting a practical limit to themaximum integration time of model RMHD P0 4 BZ.

5.3. Ohmic resistivityWe here treat only the Ohmic resistivity, though the other

non-ideal effects – ambipolar diffusion and Hall drift – areimportant in the dead zone (Bai & Stone 2013; Turner et al.2014b; Lesur et al. 2014). We have compared Eq. 11which mimics the Ohmic dissipation dependence on tem-perature with the ionization balance results from Desch &Turner (2015). Eq. 11 reproduces the transition region aroundElsasser number unity very well. However we note thatthe Ohmic resistivity is much greater inside the dead zone(Rem 1) than our computational resources allow. Fur-thermore, we consider only thermal ionization, neglecting thestellar FUV and X-ray photons and the stellar and interstellarenergetic protons. Thus the disk’s surface layers, where theseexternal fluxes ought to be absorbed, are ionized too little inthe colder parts of our models. Treating these additional fac-tors would likely affect the geometry of the magnetic fields inand around the dead zone.

5.4. Comparison to previous worksTurner et al. (2014a) showed that a magnetically supported

atmosphere could increase the near infrared excess emissionby a factor of two. However, the sublimation curve (Eq. 9)was not fully treated in that paper. In our models we find therim’s shape accurately using Eq. (9). Most of the emission inthe J and K band is then coming from a narrow region whichis not much affected by magnetic fields, except in the outerparts of the rim. As a result, we find the NIR flux increaseof about 20% at 2 microns when adding a strong and uniformvertical magnetic field. An even stronger magnetized coronawould be needed to explain the NIR excess.

5.5. Observed variabilityVariability of young star-disk systems has been the focus of

intense research for decades (Joy 1945; Carpenter et al. 2001;Morales-Calderon et al. 2011). Large sets of month-long si-multaneous optical and infrared lightcurves became availablerecently through surveys of low-mass T Tauri stars with theCoRoT and Spitzer space telescopes (Cody et al. 2014; Stauf-fer et al. 2016). A few Herbig stars have been examinedthrough imaging and/or spectroscopic time series. Sitko et al.(2012) studied the variability of the gas and dust emissionfrom the Herbig system SAO 206462, reporting spectral linechanges connected with variations in the accretion rate, andnear-infrared dust flux changes on monthly timescales withamplitudes of 10 to 20%. Our models match the observedtimescales very well, but the MRI turbulence yields some-what lower amplitudes below 10%. Wagner et al. (2015) in-vestigated the Herbig system HD 169142 at two epochs sep-arated by 10 years. They reported near-infrared variability of

45%, explaining it by a structural change in the inner dustrim. These observations suggest there may be an additionalvariability mechanism, perhaps associated with a disk wind(Bans & Konigl 2012) which could be driven by the MRI(Miyake et al. 2016). Further linking the variability to the in-ner rim shape are scattered light observations of HD 163296by Wisniewski et al. (2008). They report that shadows cast bythe inner rim vary on timescales of several years. We finallynote that occultation by vortices might be observed in highlyinclined systems. An example is AA Tau, which is inclinedabout 75 and still undergoing a strong occultation event thatbegan in 2013 (Bouvier et al. 2013). Such an event could bedue to a local thickening of the disk at a vortex like that on theinner rim in the models we present, but located further fromthe star. Highly inclined disks appear well suited to observethe rim occulting the star, especially in cases where the outerdisk is dust-depleted by radial drift or settling (Bertout 2000).Other studies relating flux dips in highly inclined systems tooccultation by dusty material include those by Alencar et al.(2010); Morales-Calderon et al. (2011); Cody et al. (2014).

Such an event could be due to a vortex located at larger diskradii, increasing locally the height of the disk, similar as thevortex at the inner rim in the models we present.

Such highly inclined disk systems might be ideally suitedto observe the star occultation by the rim (Bertout 2000), es-pecially in cases where the outer disk is dust depleted by theradial drift or settling. The studies by Alencar et al. (2010);Morales-Calderon et al. (2011); Cody et al. (2014) related dip-ping events by dust occultation which was observed in highlyinclined systems. Recently Ansdell et al. (2016) showed thatsuch events could also occur for less inclined systems.

6. SUMMARY

We have presented the first global 3-D radiation non-idealMHD models of the innermost reaches of protostellar disks,using them to investigate the dynamics and thermodynamicsof the planet-forming material. Our models include the trans-fer of the starlight into the dust and gas, where the heatingimpacts the dust sublimation and deposition and the Ohmicresistivity. The starting conditions come from axisymmetricradiation viscous hydrodynamical models of the disk arounda typical Herbig Ae star, with a radially-independent massaccretion rate of 10−8 M yr−1. Magnetic fields either withor without a net vertical flux yield magneto-rotational turbu-lence. The inner disk’s structure divides naturally into fourzones:

1. Between the star and the silicate front, the gas is turbu-lent with RMS speeds of 400 to 800 m s−1, dependingon the initial magnetic field configuration. The accre-tion stress-to-pressure ratio α is between 3 and 10%,and the turbulent magnetic field strengths are severalGauss. The gas is hotter than the silicate sublimationthreshold.

2. Lower temperatures let silicate dust exist beyond acurved front that is closest to the star in the midplaneat about 0.5 AU. Dust and strong turbulence coexist at0.5 to 0.8 AU, where temperatures are about 1000 K.The stress-to-pressure ratios are similar to those nearerthe star, but the turbulence is slightly slower, due to thelower temperatures, at 300 to 700 m s−1. High-speedcollisions should substantially limit the grains’ maxi-mum size.

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 13

3. Beyond about 0.8 AU, temperatures are low enoughand collisional ionization slow enough that the mag-netic fields decouple from the gas motions. This regionis the dead zone. From 0.8 to 1.1 AU, turbulent speedsdecline quickly with distance. Density waves propagat-ing from the turbulent region are quickly damped, leav-ing laminar gas with turbulent velocities below 1 m s−1.A local pressure maximum lies in the weakly-turbulentdead zone near 1 AU. This pressure peak is able to haltsolid particles’ radial drift.

4. Beyond 1.1 AU, well inside the dead-zone, the diskis quasi-laminar with very low turbulent speeds. Thisregion lies partly in the shadow cast by the sublimationfront.

The 3-D calculations let us investigate non-axisymmetricstability. We find Rossby wave instability develops overtimescales of 20 local orbits into a vortex located at the deadzone’s inner edge and close to the upper rim of the curved sil-icate sublimation front. The vortex moves the disk’s surfaceup and down over its orbital period.

We post-process our results using Monte Carlo radiativetransfer tools to compare against a variety of observationalconstraints:

1. Our models with strong magnetic fields have near-infrared fluxes that are 5 to 20% greater than the vis-cous version, because the magnetically-supported diskatmosphere raises the sublimation front, reprocessingmore of the starlight into wavelengths near 2 and 3 µm.Magneto-rotational turbulence could thus be a factor inthe long-standing puzzle of Herbig stars’ anomalouslylarge near-infrared excesses (Vinkovic et al. 2006; Ackeet al. 2009; Dullemond & Monnier 2010).

2. The vortex that develops near the sublimation front’shigh point locally raises the height where the starlightis absorbed, thus casting a longer shadow on the diskbeyond. The fraction of the stellar luminosity inter-cepted by the sublimation front at this stellar longitude

is increased about 10%. Such shadow-casting vorticescould potentially be related to the variability observedin scattered-light imaging of Herbig disks (Wisniewskiet al. 2008).

3. The near-infrared flux varies up to 10% due to themovements of the inner rim, on timescales of monthsto years. The regular component of the variations islarger relative to the irregular component when the vor-tex is present. Further development of this picture couldhelp in understanding why young stars with protostellardisks have such diverse optical and infrared lightcurves(Sitko et al. 2012; Cody et al. 2014).

Radiation-MHD models of the kind we have demonstratedhere open a new window for investigating protoplanetarydisks’ central regions. They are ideally suited for exploringyoung planets’ formation environment, interactions with thedisk, and orbital migration, in order to understand the originsof the close-in exoplanets.

ACKNOWLEDGMENTS

We thank John Stauffer for his valuable comments on themanuscript. We thank also Satoshi Okuzumi for helpful dis-cussions during the project. We thank Andrea Mignone forsupporting and advising us with the newest PLUTO code.Parallel computations have been performed on the Genci su-percomputer ’curie’ at the calculation center of CEA TGCCand on the zodiac supercomputer at JPL. For this work, Se-bastien Fromang and Mario Flock received funding from theEuropean Research Council under the European Union’s Sev-enth Framework Programme (FP7/2007-2013) / ERC Grantagreement nr. 258729. This research was carried out in partat the Jet Propulsion Laboratory, California Institute of Tech-nology, under a contract with the National Aeronautics andSpace Administration and with the support of the NASA Ex-oplanet Research program via grant 14XRP14 20153. Copy-right 2016 California Institute of Technology. Governmentsponsorship acknowledged.

APPENDIX

A COMPARISON ∆φ = 0.4 VS. ∆φ = 1.6

We previously reported stronger accretion stress in a smaller azimuthal domain due to stronger mean fields Flock et al. (2012b).Here we compare the two models with different azimuthal domains. In Fig. 19, left, we show the radial profiles of α in the modelsRMHD P0 4 and RMHD P1 6 averaged over the same time period of 50 inner orbits. The two models present similar radial profileswith no significant differences. However, we note that we consider here a zero-net-flux field while a net-flux field was used in ourprevious simulations (Flock et al. 2012b). In addition, we think that the short azimuthal domain still can be an issue for the modelRMHD P0 4 BZ, which uses a net vertical flux field. Here, the mean field becomes nearly as large as the turbulent component(Section 3.3) which could be a sign that the azimuthal domain is too small.

In Fig. 19, right, we take a final look at the density perturbations of model RMHD P0 4 and RMHD P1 6. Here, we calculate theFourier transform of the density along azimuth at the midplane at 0.5 AU using the same time average as before. Fig. 19, right,shows that the profiles look very similar, from which we conclude that both models represent similar turbulent characteristics.

B GENERATING INITIAL MAGNETIC FIELD CONFIGURATIONS

In the following steps we explain how to generate the initial random magnetic field. Such a field has the advantage that theMRI turbulence quickly reaches a steady-state.

1. Calculate a random vector potential for the component Ar = Arandfrfθr with r being the radius, fr being a parabolic dampingfactor proportional to (r − r0)2 to set Ar = 0 at the inner and outer radial boundary. fθ is a parabolic factor which decreasesthe potential from the midplane to the θ boundary by the factor 50 to account for the decrease of the magnetic field in thedisk corona.

2. The amplitude of the vector potential Ar is set to match a plasma beta value of 100 at the midplane.

14 Flock et al.

Figure 19. Left: Time averaged radial profile of the stress to pressure ratio α for model RMHD P0 4 (solid line) and model RMHD P1 6 (dashed line). Right:Fourier transfer of the midplane density in azimuth at the midplane at 0.5 AU for model RMHD P0 4 (red dashed line) and model RMHD P1 6 (solid line).

3. Apply a Gaussian filter (σ = 2δr) to smooth out fluctuations on grid level to obtain comparable scales as the turbulent fieldin the steady-state.

4. Calculate ∇ × A to obtain an initial Bθ and Bφ.

It is important to define the vector potential with a periodic boundary condition in φ direction. Otherwise there will be a violationof the ∇ · B = 0 condition at the φ = 0 boundary. Initially the amplitudes of Bθ and Bφ are equal. A snapshot of the initialmagnetic field is shown in Fig. 1.

For model RMHD P0 4 BZ, we add to the vector potential a constant value of Aφ. The value of Aφ is chosen to match a magneticfield strength of 100 mGauss at 1 AU, which corresponds to a plasma beta β = 2P/B2

z of β = 3.5 · 104 at 1 AU at the midplane.The resulting vertical field has a radial profile of r−1. The strength of the field corresponds to a relative high value of verticalmagnetic flux (Okuzumi et al. 2014).

C WAVE DAMPING AT THE DEAD-ZONE INNER EDGE

There are several damping mechanisms for the density waves, generated in the MRI turbulent regions and which are travelinginto the dead-zone. The most important one is the non-linear damping by shocks as soon as the wavelength is comparable to thedisk scale height (Heinemann & Papaloizou 2009b). A similar wave dissipation by weak shocking in the dead-zone was foundin global MHD simulations at the inner dead-zone edge by Faure et al. (2014b). One difference, to previous simulations is thefact that the local H/R is much smaller, meaning that waves traveling over a given radial distance are stronger damped. Anotherdifference is the radial changing thermal diffusion. The density waves travel through a region with increasing surface density, andso increased optical thickness. At the same time, in this region the MRI is switched of and there is no excitation of density wavesanymore.

The efficiency of wave damping by thermal diffusion is highest if the diffusion timescale is comparable with the typicaltimescale of the density wave. The turbulent correlation time of the MRI is roughly one tenths of the orbital period for length-scales comparable to H. At the same time, Heinemann & Papaloizou (2009b) report that the largest density waves fitting in theazimuthal domain carry most of the energy. The characteristic timescale of thermal diffusion can be estimated with the radiationdiffusion in the gas pressure dominated regime (Flaig et al. 2010):

∆tdif =

(1 +

ρε

4ER

)3κρL2

c(C1)

with the characteristic lengthscale L, which we set to the disk scale height H. In Fig. 20 we plot the midplane diffusion timescalefor this lengthscale, normalized over the dynamical timescale. The yellow bar marks the region in which the diffusion timescalebecomes comparable to the dynamical timescale, for which we expect highest damping. The zone matches the region in whichthe Reynolds stress quickly drops.

We summarize that the combination of a small H/R and a thermal diffusion timescale comparable to the dynamical timescaleat the dead-zone inner edge leads to an efficient damping of the density waves and so the Reynolds stress in the dead-zone.

D RT SETUP AND DUST HALO

We briefly summarize the setup for RADMC3D to post-process the 3D datasets. For the Monte-Carlo runs we use 210 millionphoton packages. We first transfer the grid values from the radiation MHD calculation to the RADMC grid structure. Thenwe re-calculate the thermal structure using the wavelength dependent dust opacity table. This ensures that the SED and the

3D radiation non-ideal magnetohydrodynamical simulations of the inner rim. 15

Figure 20. Thermal diffusion timescale along the midplane for the snapshot at 50 inner orbits for model RMHD P1 6. The profile follows roughly the profile of thedust density which determines the opacity and so the local optical depth. The yellow bar marks the region in which the diffusion timescale becomes comparableto the dynamical timescale (0.1 to 10 times the orbital time), for which we expect highest damping.

Figure 21. The quality factor, time averaged and mass weighted in the vertical direction. The dead-zone edge is annotated with the yellow line. The black dottedlines emphasize the 8 cell limit.

temperatures are consistent for the given dust opacity. For the dust opacity we assume the same grains size distribution as for the2D models (Appendix A (Flock et al. 2016)). All grains have the same size distribution, including those in the dust halo in frontof the dust rim. In reality, larger grains would be more likely to survive the hotter temperatures in front of the inner rim due totheir higher emission to absorption ratio (Kama et al. 2009).

We have checked that the temperature of the Monte Carlo run and the global RMHD models match exactly at the inner rim.As we neglect the gas opacity in the Monte-Carlo runs, we observe small deviations in the temperature for the very optical thinlayers of the global models. In addition, we have already shown in our previous models that the effect of the accretion heatingremains small for these model parameters (Flock et al. 2016).

We want also to discuss again the dust halo which appears in our models (Flock et al. 2016). The main reason for this halois the difference between the gas and dust temperatures in the optically thin environment. For the case T > Tev, the dust startsto evaporate, however pure gas alone would lead to a temperature below the evaporation temperature. The solution is that a tinyamount of dust condenses to balance the temperature drop. The result is a small dust halo in which the temperature is close tothe evaporation temperature. A similar result was found by Kama et al. (2009), however in their model, larger dust grains areresponsible as they have a larger emission to absorption ratio and so survive in front of the rim. In our models, the gas componenthas a larger ε value which leads to a temperature which is cooler than dust in an optical thin environment.

E MRI QUALITY FACTOR

Following the work by Noble et al. (2010) and Sorathia et al. (2011) we determine the quality factor Q which shows the numberof grid cells per fastest MRI growing mode. The quality factor Qφ for the azimuthal field is defined as

Qφ = 2π

√1615|Bφ|√4πρ

1Ω

1r∆φ

. (E1)

In Fig. 21 we plot the quality factor for the models RMHD P0 4 and RMHD P0 4 BZ. Space and time averaging follows the samestrategy as in Section 3.2 and 3.3. The plot shows that both models resolve very well the MRI in the region with high ionizationwith 16 or more grid cells per fastest-growing MRI wavelength. Model RMHD P0 4 BZ shows a even higher quality factor due tothe stronger field (Figs. 6 and 7).

16 Flock et al.

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